5. Prove that for any positive integer n>=5, there exists a graph with n vertices, all of which have degree 4. Use induction.

6. Suppose G is a graph with no cycles of length 3, where every vertex has degree k.
(a) Show that G has at least 2k vertices. All is needed is a picture.
(b) If G has exactly 2k vertices, what does it look like? Make sure your answer works for every value of k = 1,2,3...

7. A bridge is an edge in a connected graph whose removal disconnects the graph.
(a) Show that a connected graph with all degrees even cannot have a bridge. Hint: Suppose it did have a bridge. What happens when you remove it?
(b) On the other hand, for every odd number n, construct a connected graph G where every vertex has degree n and G has a bridge.
Draw it for n = 1 (easy) and n = 3 (harder). Study the n = 3 case and make something similar work for larger odd n.

8. There are 50 scientists at a conference and each of them is acquainted with at least 25 of other others. Show that there are four of them who can be seated at a round table so that each of them has two acquaintances for neighbors.
This can easily be solved using a picture.